Adeles and Tamagawa numbers
Cours de Jean-Pierre Serre, no. 1 (1981) , 204 p.

Sommaire

Historyp. 1
Siegel formulap. 3
Tamagawap. 8
I - Integration on $p$-adic manifolds pagep. 9
Notationp. 9
Measure attached to a form $\omega$ p. 10
Rational summationp. 15
Igusa p. 16
Vector bundlesp. 17
Smooth schemesp. 18
Number of points of classical and exceptional groupsp. 20
Decomposition of a measurep. 24
Connection with densitiesp. 28
Digression : Liftable solutions p. 31
Smooth casep. 32
Real casep. 37
Use of resolution of singularitiesp. 40
Oesterlé : hypersurfaces p. 41
Łojasiewicz inequality p. 48
Oesterlé : general casep. 49
II - Adelesp. 55
Historyp. 55
Definitionp. 55
$\mathbb{A}_K/K$ compactp. 58
Haar measure on $\mathbb{A}_K$ p. 61
Characters and duality p. 64
Haar measure for dual groupsp. 66
Compatible measures with respect to duality p. 67
Adelic integration and heuristic formulas (Goldbach,...) p. 69
Adelic points of algebraic varieties p. 75
Properties of the functor $V\mapsto V(\mathbb{A}_K)$ p. 78
Restriction of scalarsp. 84
Algebraic groups and adelic pointsp. 88
Abelian varieties (S. Bloch) p. 88
Weak approximationp. 90
Strong approximationp. 91
Adeles, classes and generap. 99
Tensors pagep. 102
Vector bundlesp. 104
Adelic measuresp. 106
Convergent casep. 108
Algebraic groupsp. 111
Torip. 114
Convergencep. 117
Tamagawa number (semi-simple case)p. 122
Theorem of Ono – Weil conjecturep. 123
Tamagawa number (reductive case)p. 126
$\tau(\mathbb{G}_m)=1$ p. 131
Tori (Ono)p. 133
$\tau(PGL_n)=n \tau(SL_n)$p. 135
Tamagawa $\leftrightarrow$ Siegel (mass formula) p. 136
Tamagawa $\leftrightarrow$ Siegel (the two-groups game)p. 145
Correction to Ono's theory p. 146
Positive definite quadratic forms over $\mathbb{Q}$ p. 147
Proof that $\tau(SO_n)=2$ for $n=2,3,4$p. 152
Proof of Siegel's formula p. 156
Proof ($m\geq 5$)p. 163
Proof ($m=4$)p. 165
Proof ($m=2$)p. 166
Proof ($m=3$) p. 169
Remarks on Siegel's proofp. 172
Application to modular forms p. 176
III - $SL_n$p. 180
The Minkowski-Hlawka theoremp. 180
Proof p. 184

@book{CJPS_1981__1_,
     author = {Serre, Jean-Pierre},
     title = {Adeles and {Tamagawa} numbers},
     series = {Cours de Jean-Pierre Serre},
     publisher = {Harvard},
     number = {1},
     year = {1981},
     language = {en},
     url = {http://www.numdam.org/item/CJPS_1981__1_/}
}
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AU  - Serre, Jean-Pierre
TI  - Adeles and Tamagawa numbers
T3  - Cours de Jean-Pierre Serre
PY  - 1981
DA  - 1981///
IS  - 1
PB  - Harvard
UR  - http://www.numdam.org/item/CJPS_1981__1_/
LA  - en
ID  - CJPS_1981__1_
ER  - 
Serre, Jean-Pierre. Adeles and Tamagawa numbers. Cours de Jean-Pierre Serre, no. 1 (1981), 204 p. http://numdam.org/item/CJPS_1981__1_/